A new theorem on exponential stability of periodic evolution families on Banach spaces

Abstract

We consider a mild solution v_f (·, 0) of a well-posed inhomogeneous Cauchy problem v̇(t) = A(t)v(t) + ƒ(t), v(0) = 0 on a complex Banach space X, where A(·) is a 1-periodic operator-valued function. We prove that if vƒ (·, 0) belongs to AP0 (ℝ₊, X) for each ƒ ∈ AP0(ℝ₊, X) then for each x ∈ X the solution of the well-posed Cauchy problem u̇(t) = A(t)v(t), u(0) = x is uniformly exponentially stable. The converse statement is also true. Details about the space AP0(ℝ₊, X) are given in the section 1, below. Our approach is based on the spectral theory of evolution semigroups.